arXiv subjects

Jin Ma

Publications and source records attributed to Jin Ma.

At least 109 records · Page 6Linked to original sources

arXiv2018

Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We develop first a local theory of classical solutions and define then viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative one using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. When the diffusion coefficient is semi-linear (but the drift can be fully nonlinear), we establish a complete theory, including global existence and comparison principle. Our methodology relies heavily on the method of characteristics.

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arXiv2017

Kyle-Back Equilibrium Models and Linear Conditional Mean-field SDEs

In this paper we study the Kyle-Back strategic insider trading equilibrium model in which the insider has an instantaneous information on an asset, assumed to follow an Ornstein-Uhlenback-type dynamics that allows possible influence by the market price. Such a model exhibits some further interplay between insider's information and the market price, and it is the first time being put into a rigorous mathematical framework of the recently developed {\it conditional mean-field} stochastic differential equation (CMFSDEs). With the help of the "reference probability measure" concept in filtering theory, we shall first prove a general well-posedness result for a class of linear CMFSDEs, which is new in the literature of both filtering theory and mean-field SDEs, and will be the foundation for the underlying strategic equilibrium model. Assuming some further Gaussian structures of the model, we find a closed form of optimal intensity of trading strategy as well as the dynamic pricing rules. We shall also substantiate the well-posedness of the resulting optimal closed-loop system, whence the existence of Kyle-Back equilibrium. Our result recovers many existing results as special cases.

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arXiv2017

A Mean-field Stochastic Control Problem with Partial Observations

In this paper we are interested in a new type of {\it mean-field}, non-Markovian stochastic control problems with partial observations. More precisely, we assume that the coefficients of the controlled dynamics depend not only on the paths of the state, but also on the conditional law of the state, given the observation to date. Our problem is strongly motivated by the recent study of the mean field games and the related McKean-Vlasov stochastic control problem, but with added aspects of path-dependence and partial observation. We shall first investigate the well-posedness of the state-observation dynamics, with combined reference probability measure arguments in nonlinear filtering theory and the Schauder fixed point theorem. We then study the stochastic control problem with a partially observable system in which the conditional law appears nonlinearly in both the coefficients of the system and cost function. As a consequence the control problem is intrinsically "time-inconsistent", and we prove that the Pontryagin Stochastic Maximum Principle holds in this case and characterize the adjoint equations, which turn out to be a new form of mean-field type BSDEs.

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arXiv2016

Fast Stability Scanning for Future Grid Scenario Analysis

Future grid scenario analysis requires a major departure from conventional power system planning, where only a handful of most critical conditions is typically analyzed. To capture the inter-seasonal variations in renewable generation of a future grid scenario necessitates the use of computationally intensive time-series analysis. In this paper, we propose a planning framework for fast stability scanning of future grid scenarios using a novel feature selection algorithm and a novel self-adaptive PSO-k-means clustering algorithm. To achieve the computational speed-up, the stability analysis is performed only on small number of representative cluster centroids instead of on the full set of operating conditions. As a case study, we perform small-signal stability and steady-state voltage stability scanning of a simplified model of the Australian National Electricity Market with significant penetration of renewable generation. The simulation results show the effectiveness of the proposed approach. Compared to an exhaustive time series scanning, the proposed framework reduced the computational burden up to ten times, with an acceptable level of accuracy.

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arXiv2016

Optimal Dividend and Investment Problems under Sparre Andersen Model

In this paper we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a "renewal" process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution} to the associated HJB equation on a cylindrical domain on which the problem is well-defined.

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arXiv2016

Dynamic Approaches for Some Time Inconsistent Problems

In this paper we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept of time-consistency as well as any method that is based on Pontryagin's Maximum Principle. The fundamental obstacle is the dilemma of having to invoke the {\it Dynamic Programming Principle} (DPP) in a time-inconsistent setting, which is contradictory in nature. The main contribution of this work is the introduction of the idea of the "dynamic utility" under which the original time inconsistent problem (under the fixed utility) becomes a time consistent one. As a benchmark model, we shall consider a stochastic controlled problem with multidimensional backward SDE dynamics, which covers many existing time-inconsistent problems in the literature as special cases, and we argue that the time inconsistency is essentially equivalent to the lack of {\it comparison principle}. We shall propose three approaches aiming at reviving the DPP in this setting: the duality approach, the dynamic utility approach, and the master equation approach. Unlike the game approach in many existing works in continuous time models, all our approaches produce the same value as the original static problem.

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arXiv2015

On well-posedness of forward-backward SDEs-A unified approach

In this paper, we study the well-posedness of the Forward-Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to address some fundamental longstanding problems for non-Markovian FBSDEs. An important device is a decoupling random field that is regular (uniformly Lipschitz in its spatial variable). We show that the regulariy of such decoupling field is closely related to the bounded solution to an associated characteristic BSDE, a backward stochastic Riccati-type equation with superlinear growth in both components $Y$ and $Z$. We establish various sufficient conditions for the well-posedness of an ODE that dominates the characteristic BSDE, which leads to the existence of the desired regular decoupling random field, whence the solvability of the original FBSDE. A synthetic analysis of the solvability is given, as a "User's Guide," for a large class of FBSDEs that are not covered by the existing methods. Some of them have important implications in applications.

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arXiv2015

Stochastic differential equations driven by fractional Brownian motion and Poisson point process

In this paper, we study a class of stochastic differential equations with additive noise that contains a fractional Brownian motion (fBM) and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which the long term dependence between the claim payment and the past history of liability becomes the main focus. We establish some new fractional calculus on the fractional Wiener-Poisson space, from which we define the weak solution of the SDE and prove its existence and uniqueness. Using an extended form of Krylov-type estimate for the combined noise of fBM and compound Poisson, we prove the existence of the strong solution, along the lines of Gyöngy and Pardoux (Probab. Theory Related Fields 94 (1993) 413-425). Our result in particular extends the one by Mishura and Nualart (Statist. Probab. Lett. 70 (2004) 253-261).

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arXiv2014

Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation

This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao and Liu \cite{GL}, this extends the corresponding results collected in Freidlin and Wentzell \cite{FreidlinWentzell}. However, we use a different line of argument, adapting the PDE method of Fleming \cite{Fleming} and Evans and Ishii \cite{EvansIshii} to the path-dependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path-dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics.

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arXiv2014

Dynamic Equilibrium Limit Order Book Model and Optimal Execution Problem

In this paper we propose a dynamic model of Limit Order Book (LOB). The main feature of our model is that the shape of the LOB is determined endogenously by an expected utility function via a competitive equilibrium argument. Assuming zero resilience, the resulting equilibrium density of the LOB is random, nonlinear, and time inhomogeneous. Consequently, the liquidity cost can be defined dynamically in a natural way. We next study an optimal execution problem in our model. We verify that the value function satisfies the Dynamic Programming Principle, and is a viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation which is in the form of an integro-partial-differential quasi-variational inequality. We also prove the existence and analyze the structure of the optimal strategy via a verification theorem argument, assuming that the PDE has a classical solution.

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arXiv2013

Pathwise Taylor Expansions for Random Fields on Multiple Dimensional Paths

In this paper we establish the pathwise Taylor expansions for random fields that are "regular" in the spirit of Dupire's path-derivatives \cite{Dupire}. Our result is motivated by but extends the recent result of Buckdahn-Bulla-Ma \cite{BBM}, when translated into the language of pathwise calculus. We show that with such a language the pathwise Taylor expansion can be naturally carried out to any order and for any dimension, and it coincides with the existing results when reduced to these special settings. More importantly, the expansion can be both "forward" and "backward" (i.e., the temporal increments can be both positive and negative), and the remainder is estimated in a pathwise manner. This result will be the main building block for our new notion of viscosity solution to forward path-dependent PDEs corresponding to (forward) stochastic PDEs in our accompanying paper \cite{BMZ}.

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arXiv2012

Optimal Portfolio Selection under Concave Price Impact

In this paper we study an optimal portfolio selection problem under instantaneous price impact. Based on some empirical analysis in the literature, we model such impact as a concave function of the trading size when the trading size is small. The price impact can be thought of as either a liquidity cost or a transaction cost, but the concavity nature of the cost leads to some fundamental difference from those in the existing literature. We show that the problem can be reduced to an impulse control problem, but without fixed cost, and that the value function is a viscosity solution to a special type of Quasi-Variational Inequality (QVI). We also prove directly (without using the solution to the QVI) that the optimal strategy exists and more importantly, despite the absence of a fixed cost, it is still in a "piecewise constant" form, reflecting a more practical perspective.

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arXiv2010

Backward SDEs with constrained jumps and quasi-variational inequalities

We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence, this suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs.

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arXiv2010

Pathwise Taylor Expansions for Itô Random Fields

In this paper we study the {\it pathwise stochastic Taylor expansion}, in the sense of our previous work \cite{Buckdahn_Ma_02}, for a class of Itô-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work \cite{Buckdahn_Ma_02}, our new expansion can be defined around any random time-space point $(\t,ξ)$, where the temporal component $\t$ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point $(\t,ξ)$. As an application, we show how this new form of pathwise Taylor expansion could lead to a different treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence lead to a definition of the {\it stochastic viscosity solution} for such SPDEs, which is new in the literature.

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arXiv2009

On Quadratic g-Evaluations/Expectations and Related Analysis

In this paper we extend the notion of g-evaluation, in particular g-expectation, to the case where the generator g is allowed to have a quadratic growth. We show that some important properties of the g-expectations, including a representation theorem between the generator and the corresponding g-expectation, and consequently the reverse comparison theorem of quadratic BSDEs as well as the Jensen inequality, remain true in the quadratic case. Our main results also include a Doob-Meyer type decomposition, the optional sampling theorem, and the up-crossing inequality. The results of this paper are important in the further development of the general quadratic nonlinear expectations.

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arXiv2009

Stochastic differential equations driven by fractional Brownian motions

In this paper, we study the existence and uniqueness of a class of stochastic differential equations driven by fractional Brownian motions with arbitrary Hurst parameter $H\in (0,1)$. In particular, the stochastic integrals appearing in the equations are defined in the Skorokhod sense on fractional Wiener spaces, and the coefficients are allowed to be random and even anticipating. The main technique used in this work is an adaptation of the anticipating Girsanov transformation of Buckdahn [Mem. Amer. Math. Soc. 111 (1994)] for the Brownian motion case. By extending a fundamental theorem of Kusuoka [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 567--597] using fractional calculus, we are able to prove that the anticipating Girsanov transformation holds for the fractional Brownian motion case as well. We then use this result to prove the well-posedness of the SDE.

math.ST↗

arXiv2009

Optimal reinsurance/investment problems for general insurance models

In this paper the utility optimization problem for a general insurance model is studied. The reserve process of the insurance company is described by a stochastic differential equation driven by a Brownian motion and a Poisson random measure, representing the randomness from the financial market and the insurance claims, respectively. The random safety loading and stochastic interest rates are allowed in the model so that the reserve process is non-Markovian in general. The insurance company can manage the reserves through both portfolios of the investment and a reinsurance policy to optimize a certain utility function, defined in a generic way. The main feature of the problem lies in the intrinsic constraint on the part of reinsurance policy, which is only proportional to the claim-size instead of the current level of reserve, and hence it is quite different from the optimal investment/consumption problem with constraints in finance. Necessary and sufficient conditions for both well posedness and solvability will be given by modifying the ``duality method'' in finance and with the help of the solvability of a special type of backward stochastic differential equations.

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arXiv2009

Weak solutions for forward--backward SDEs--a martingale problem approach

In this paper, we propose a new notion of Forward--Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the forward--backward stochastic differential equations (FBSDEs). The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of an FBSDE. We first prove a general sufficient condition for the existence of the solution to the FBMP. In the Markovian case with uniformly continuous coefficients, we show that the weak solution to the FBSDE (or equivalently, the solution to the FBMP) does exist. Moreover, we prove that the uniqueness of the FBMP (whence the uniqueness of the weak solution) is determined by the uniqueness of the viscosity solution of the corresponding quasilinear PDE.

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